Discussions, incl. FW Lawvere, G Khatcherian & M Wright

0:00 I, you know, I did, I just like me that this, this business, the fact that he uses this term reification repeatedly, as if it was a, as if he diagnosed a precise philosophical weakness in, it's incredible to me because really what he's criticizing is, is just, is actually not reification, but dialectical reflection. I mean, he says all structuralist mathematics, you know, you are a representative of structuralist mathematics, all structuralist mathematics, whether the set theoretic or the category theoretic, is congenitally unable to deal with process as subject matter except indirectly through its ratification. Well, of course it isn't, but then nobody has ever claimed it is. Of course you deal with process indirectly. In the sense, what does rarefication mean here? I mean, what on earth does rarefication mean? But what can Lenin say about it, you know? It's... We must. We must. I mean, there's no... There's no other way of doing it. Reification here is just reflection in thought. What does reification mean here? It's a representation in a form suited to the real laws of motion of a subject matter. This is objective dialectics, not Wolffian metaphysics. He's forgotten that objects are objects together with their relations. And then he says, let me be more explicit, a functor may represent motion, but only in highly reified form. Anything which represents does so in a rarefied form. It is not itself motion or process of any kind. Of course it isn't, and nobody except probably Finkelstein was ever so confused as to claim it was itself process.

2:30 I'm very surprised indeed that I got this, because I was surprised. I'm glad he liked your lecture so much. But I think I'm very confused about this. The first part reminded me of, I've never seen this, but Chanuel assures me that there are three very circles there are of papers. Joyce on Shakespeare. Yeah, yeah, yeah. The program was Interpreting Mathematics. It rose from French Structuralism. Yeah, it is a bit like that. One of the great people is Hausdorff, for example. I wasn't aware that they were introspective about key. That's very strange. I was very surprised with you. I just thought you'd be a bit disappointed in that. May I suggest, because I think it's going to be a bit too late now to get lunch in the pub, why don't I just rustle up some... Would you like some paella or some risotto? And we could go and sit out in the garden and eat it. Yeah, so we could take the table outside and I'll just pop down and get some wine. Well, we can go out when Gerry gets in the car this evening. Because unfortunately they'll only be open for about another half an hour now, so there's no point in walking out there. I'll just go and do this. Oh, meanwhile, there was one other thing. I wanted to ask about many things, but I've got to organize them in a little time. One is about some things in this book, but there's a more straightforward one. You mentioned Lambic, Joachim. What do you think of his paper about you, the Heraclitus and modern mathematics? The paper is called Heraclitus and Modern Mathematics, which is all about Bilbao here.

5:00 You don't know this paper? I knew he had some. It's been around for quite a long time. I think it was published in the PSA for about 1978. I'll get a copy of it. Yeah, I think it's, I think that a lot of, I think that some of the misunderstandings in John Bell, John, that last chapter of John Bell's book, you know, about relative, relativization of mathematical truth, you know, truth, everything is local in some sense that actually gets at the, the very nature of mathematics as a reflection of the world, which I, I think is. It has to be confused, very confused. I think some of it comes from this, and I'd be interested if you know what you think of it. Yeah, I must be honest, it's been a long time since I read that book. Well, I thought hardly anything was known about Heraclitus. Well, and what was that? I don't know. Well, he actually had quite a reason to do that. Yeah. In other words, the basic point, according to this account, Heraclitus had both aspects that I can picture.

7:30 As people think of him as, do they? As a contrast, as a contributor, that proves that this is Queen Heraclitus. I have no knowledge of Greek philosophy. I mean, I've read texts. But I don't have any sort of scholarly knowledge of the arguments about who really thought what, I mean, I don't even know any. But Heraclitus has always seemed to me to be one of the major sources of, major sources for objective idealism, one form or another. But again, the idea is that everything is in such a flux that we can never understand it. Yes. But you see, this is exactly where Finkelstein's ideas come. Yeah, yeah. And I mean, yeah, so I absolutely know, I mean, it's a case of speaking with as much exactness as the subject matter permits of, as Aristotle would say. It's not something which is like a proof in homotopics, too. No, it's perfectly clear.

10:00 I'm perfectly correct. I think the, it's the old, and in fact it's precisely this, a resurrection of precisely this, you know, this Heraclitus, well, not perhaps the view of the historical Heraclitus, but that, that view that was attributed to him by Plato when he set him up as the straw man, yeah, was, yeah, yeah, is precisely, I think, where, you know, Finkelstein is, when he... He says that, well, you know, we've tried to go from products to bundles to the factories inside larger products and then found out that they weren't objects at all. You know, constantly we've tried to go from objects to transformation in search of true objects. Well, every object is already its own transformation. So let us start from there and let us try and think what it means to have a quantum arithmetic of partially defined mappings and try and do without and try and explain how exponentiation collapses to product and it's all to be explained in terms of the properties of the matrices just have one zero instead of one on each diagonal. It's all kind of quite crazy. Because you can say, well, if you knew any category theory, you could just take the endo maps in the function space and see that that is just nonsense, but then you would have understood what it is about the Bill of Your Principles and anti-science that you're missing. I do, I do. And I mean, that was exactly what I hadn't understood until yesterday when, or the day before, when you explained to me what constants, what randomness, you know, the relationship between the absolutely random motion, the kind of category of absolutely random motions, and the category of absolutely constant immobility, or whatever you want to call it.

12:30 Whatever you, however you want to term it, the constant, the category of absolute constancy was and how it reflected ideas about being and becoming. As long as you calculate the cardinality, it's important first. Alright. And then you can, you can approximate the states on the right and left, but you can just that. There's nothing wrong about that. Of course, it's not the whole story of the space either. Yeah, yeah. As set theorists tend to think that it is the whole story of the space. Well, they tend to think, yes. Yeah, it is. But I mean, to say that... Well, that you would start with fixed determinant objects in... But to say that it's wrong, somehow, in other words, the progress of knowledge is a kind of comedy of errors. You go to the next stage, and in order to be wrong again, it's a kind of... Oh, exactly. ...facility. Absolutely, yes. Well, indeed, again, you see it in Penolstein, the whole construction collapses to its base, and we have to start all over again with a completely new idea, which is that becoming is prior to being. And really, it's an old, old... ...pattern in the, well, ever since Greek philosophy. Now, I mean, historically, I don't know what you're telling me. I may very well be right, but in fact, there wasn't, the real historical Heraclitus wasn't responsible for this.

15:00 That's why I'm a little bit worried about some of John's formulations in that paper, in the end of the topology section. Of course, it's the same thing. It is the same thing. And in fact, I'll tell you why I was particularly alerted to what it was to worry about in that. Because... Finkelstein, you obviously know, I've been seeing a lot of the guy, he is a friend, so I don't want you to think I'm bad mouthing him as a person, he asked me, which shows he must have been in a bad way, to tell him a bit about toposys, and of course I did say, well, you know, you might as well ask a monkey to explain a screw wrench, but here it goes, so I explained what little I knew. So, as he read that last chapter of John's, he seized on it with glee, saying, aha, now here is a man who thinks just as I do, and, this is the blue one, not appropriate actually, it should be bound, it should be red, this is all your place again, it should be bright red, I must refile those, the western appropriate colour I could have chosen. Oh, yeah, a couple of things about metric spaces, generalised logic, I wanted to ask you, but as soon as he, Finkelstein, had finished reading the chapter, you know, the last chapter of John's book, I mean, I don't think he, well, he seems to me to be quite a competent mathematical physicist in his work, he immediately said to me, oh, yes, yes, this is exactly, you know, all he needs to do is to go one stage further, and he would see.

17:30 But if you go to the case of local set theory, quantum set theory, which already has locality built in, where was it, this strange idea? Pretty crude. Well, it's useful because it means I can keep things together when all the papers that are relevant to a particular subject, like yours, and kind of refer to them together. Yes, I mean, he's got these kind of very crude intuitions about how to proceed. Now, our work on quantum set theory is stimulated by the work of Scott and Solovey on random sets and by Takauchi on quantum sets. When we deal with the quantum system, We replace the scope of the classical indefinite objects, he's got an explanation of what indefinite objects are supposed to be, with its commutative function algebra and its distributive subset lattice by a linear space or more generally by a module over a ring with a non-commutative operator algebra and a non-distributive subspace lattice, so that the main mathematical difference between the indefinite set theory and other set theories such as fuzzy, hazy, I've never heard. There's something called Dodson. Random, Scott and Sol, are they? And the quantum set theory of Tachyutti is how the hierarchy of sets is generated. Our theory, yeah, it's okay, it should be. It needs to do that for about 20 minutes, 15 minutes, and then it's ready to eat, yeah.

20:00 Yes, indefinite set theory uses the tensor product to do it. I must admit, this I did find quite an interesting remark, which I wish he could. To explain further what he means, indefinite set theory should be thought of as using the tensor product to generate the algebra of projections inductively, concurrently with the generation of the sets. While Takeuchi quantum sets and Scott-Solovey sets use a quite arbitrary fixed algebra of projections already chosen at the start. And he, as I say, got very excited by John's chapter on local set theory. And he started saying to me that, oh, yes, this is just like my ideas about the relativity of topology and dynamics, that everything, you know, topology and dynamics are, you know, dynamics is primal logic, in some sense, and dynamics and topology are already relative. And each, well, he said, we shall speak of a trio of relativities, each enlarged by the next, giving an increasingly faithful account of the relations of different control systems, he calls them, CSs. The first is classical relativity including Galileo's and Einstein's. It assumes two different control systems have the same possibilities except for a relabeling classical transformation by the control system, so x prime is equal to function of x. The second is CQ, classical quantum relativity, which is Bohr's. And also, he thinks, Davis, the man who did this work on complete Boolean algebras of projections representing real numbers as self-adjoint operators on Hilbert space, and saying, well, that the quantization of a theory just is an application of a Boolean-valued model to it. It assumes two different control systems see the same system and that for each control system the possibilities for the system are defined by a complete orthonormal basis in a Hilbert space associated with the system and the control system.

22:30 This allows each control system to see new possibilities related to the old by unitary transformation. CQ, classical quantum, Copenhagen quantum, transformation of the control system. Now our relativity is third relativity, true quantum relativity. It allows for the fact that two different control systems see different worlds since each sees the other but not itself. And this is just sheer mysticism. But each sees the other and not itself. A quantum transformation is generally not a unitary transformation since the two control systems may have different numbers of possibilities in their basis. An analyzed notion of possibility and the relativity of topology and dynamics is this Q-relativity I mean I'm not suggesting that this is deep I'm just saying that it helps me understand where his ideas come from and I think they are in front of direct line of descent from these Heraclitian ideas and the second what you call the second era of anti-science. And it results in these strange applications of Grassman algebras, which are supposed to be in some sense, I don't understand why they're, both Grassman algebras and Clifford algebras, they're in some sense, they're algebras of operators, and they're supposed to be... The actual mathematics was rather murky. Very, very murky. But then of course he thinks that there is something fundamentally wrong with mathematics because it hasn't taken account of... Therefore it should not be done carefully. Well, that's a very bad conclusion, I think. I know, I know. It's difficult for somebody without... But this is a common mode of reasoning among physicists themselves. But since our descriptions are only approximate, therefore the mathematics is also only approximately done.

25:00 Things which do obey the principles. It is about the framework we've had for 60 years which makes possible all these non-sensors. For quantum, yes. There are no beings. No, but I mean, in actual practice, mathematics is not done carefully. I'm sorry, what you're saying about the framework we've had for 60 years, I thought you were talking about quantum theory. Yes. Oh, you are? No, but not the actual social practice of physicists. Right. Yes, yes, I mean, the underlying. Which means that since our descriptions are only in rocks, or the proofs are also in rocks. Yeah. He comes absolutely clean. Total lack of coherence as well as total lack of rational... Absolutely. Absolutely. I mean, he actually says, I'm going to show you a perfect illustration of this. We cannot... I mean, he wants to quantize. Remember, everything has to be quantized. Every... I would like to see an actual, careful definition or explanation of why Osmond and Kuiper-Dalvin are the same thing in some context. No, it is terribly unclear. I thought when I read this stuff the first time that it was unclear because I just didn't know enough mathematics to be able to understand it, but now having gone through it carefully with good mathematical physicists, people like Saunders, from whom I've learnt a lot... But I didn't know, even six months ago. I now know that wasn't the case. It was that it really is intrinsically confused. It's not just that I didn't know enough to understand what he was getting at, which is what I did think at first. In this business, we cannot straightforwardly quantize set equality.

27:30 It is impossible to know all the properties of Q-objects. It is therefore impossible to know the equality of two Q-objects. It is not clear that where equality holds, there will be two objects rather than one. So we avoid the problem by basing our Q-set theory on the bracket operation rather than on equality relations or membership. And this has become an important maxim for us. Operations are more basic than predicates. But you see, I think that source of that confusion is a kind of Heraclitus, it does go back to, well, beyond Sally Heraclitus, not the real Heraclitus, perhaps. So apparently Plato's dialogue is not called Heraclitus, it's called Cretulis. Oh, yes, Cretulis. And is he the guy who was in the middle? The kind of process metaphysics aspect, yes. No beings, only process, only eternal blocks. Oh, with both, you mean. There was another seaman, not so famous, but it played a crucial role. And who was that? That's just a kind of mnemonic for remembering.

30:00 Oh, yes, I guess it was Leibniz. Well, Leibniz is Heraclitus and Volf is Cretalus. No, it's Plato. But the guy in between was... I don't know. I'm afraid I don't know enough history of the 18th century. In fact, in an insane field craft, you know, the same as that of Virgin Friends, you know. Except they had the actual power to play that. No, no, no. Yeah, I'm not up in this. But it was a very essential, very essential move, a very reasonable start in finding this, in some ways. Well, yes, it's obviously a tremendous intellect. I mean, it's very... Someone who is very efficient in the sense that you want to deal with the world as a whole and all its relationships. Exactly, yes. Well, his wrestling with combinatorics was very impressive. He didn't get the quantifier, but he got very close. And he got, well, not least the calculus. All of these things are being in the process of development and expansion. Yes, even though he did obviously theologize that and turned it into an idealism. Well, and in 17th century England, too, of course. I mean, Newton was almost lost to Cambridge because he denied the Trinity.

32:30 Yes, the idea that metaphysics is a bad thing. Yes, yes, because, I mean, Marx certainly never, is never dismissive when he speaks about Aristotle. He's never, you know, he's clearly aware that there's very powerful scientific intelligence at work there, even though it's such a closure. Right, I think then if you, you like paella, don't you? Good, good, good. I'm going to have myself some risotto, and then we'll take this outside. We've got quite enough there for purpose. In fact, do you want to just open the door on the loger and let in some fresh air, it's just the... Yeah, that's right, you can take the table outside. Oh, hang on, it's locked. Wait a minute, I need to get my key. Yeah, I just finished doing... I did that myself. Built that with my own hands, I'm proud to say. It took me about nine months to do. I was just an old potting shed out there. I wouldn't like to pretend that I am, but I just decided, you know, I had to get this house into shape if I was going to sell it. I taught myself bricklaying to do it, and I got a lot of satisfaction out of it. It's a very satisfying thing to do.

35:00 I don't think I'd like to have to make my living at it. I don't think I'd be very good at it, but it's satisfying to be able to do it. Well, I was thinking that what I must do is get the keys, oh, it's here now. There we are, out here in the shade. Yeah, working good enough, anyway.

37:30 No, I must admit, even lying in the sun, I'm not particularly a great sun worshipper. Oh, yeah, I like it up to a point, but I guess it's coming from a cold, damn foggy island you get used to. You acclimatise to rain and cold and this is just about ready. I'm going to go and get the wine, but that's okay because we're going out to a pub this evening anyway. Did you say you wanted some vitamins? Yeah, I should be about done. And I'm sorry, I'm not one of nature's cooks, Bill, I'm afraid. This is not cordon bleu. This is out of a packet, as you obviously know, but a proper meal tonight. I'll do it myself. I imagine, I would guess that Fatima's a very good cook, isn't she? Oh, she is. As is my daughter. My daughter's been an excellent cook since she was eight years old. She used to prepare dinner parties for us. What does she do now, Bill? Where is she now? She's much with you about that. She's more or less just working at odd jobs. She doesn't have the time to start at the university. She follows her parents. That's right. Okay.

40:00 Right. Take that outside. Yeah, that'll be ready in about five minutes. I think I might take some coke. Would you like some cold milk? Yes, thanks. It's nice and cold. It's skimmed, too, so it's not... You're on food. No, it's still boiling. It won't be ready for another five minutes. I'll just come and sit with you for a bit. Actually, I'll turn it down on the burner so it doesn't...

42:30 Yeah, I just laid this patio, too. Well, I'm not sure you'd call it patio at the end of the garden, but... Because that was all kind of like a human, I think I'm going to plant, I'm going to have roses all around it. Let's back and have, yeah, yeah, just have a whole rose garden. Yeah, I was very lucky and, well, yes, but of course it has a lot of memories too. I mean, some of them, obviously the last few years, quite unhappy ones. But because, so yeah, it is a nice house, but it's really a bit big for one person. And the other thing is that this is an area where... You're trying to dispose of some of your books, is that it? Not likely. Well, it seems just the right size. Yeah, it probably needs another couple of stories actually to be the right size. You haven't seen the garage yet. There must be about another three or four thousand of them in there. Well, about two and a half thousand. Including all the Marxist classics. I just wanted to... Yes, you see, it does worry me the formulations that John adopts in this, in this, the wider significance of topos theory. Yeah, sure, you know, the set concept has turned out to be radically underdetermined, but he reads into this, he actually reads an ontological significance into this radical underdetermination. It's as if radically underdetermined, relativized local set theory is actually getting us the structure of the world, because the structure of the world really is Heraclitian in this sense. I think that's, and these extended analogies with the theory of relativity. Yeah, yeah, yes, sure. I think they're a little bit, I'm a little bit uncertain. I mean, of course, I don't know enough of the mathematical background to be sure. But he actually mentions the day very specifically as an instance of relativization of mathematical concepts, the idea of reference frame in mathematics.

45:00 This is the Davis who did this work on representing the reels as self-adjoint linear operators that is used in, well, Takeyuti's, the basis of this theory of Takeyuti's that John was talking about when David Holdsworth, after David Holdsworth gave his paper at the workshop, which he didn't, which he, Takeyuti, didn't push any further, but he, Takeyuti's work was basically a fake. The stuff about quantiles, you know. Essentially all that they get is a certain commutative portion of the commutative algebra. Yeah. So, I mean, if mechanics says anything valid about it at all, it's that there is this really non-commutative algebra, things like that. But all they look at is this commutative fragment, which is in reality already 1930. I mean, if you were to dress it up and make it more obscure by using topo theory, it doesn't fit. Yes, that's why I think that the route that, and if you believe that this project is worth pursuing at all, then the way that people like David Holdsworth go about it is a much more honest way of pursuing it, because they are really trying to get the construction of, you know, they really are trying to get a quantum set here out of something more fundamental. I know his quantum topos doesn't seem to work as well, but at least he thinks it's fixed.

47:30 Very nice Chinese lady, former Red Guard. Oh, yes. Her thesis actually probably has more to say. She uses the category of convex set. Mm-hmm. Mm-hmm. You know, it's a tensor product. If you just look for the mathematical structure, which is really common, it's a convex set. But one has it that this convex set is the set of all probability measures on a measurable space or on a topological space or something. But that sort of thing has a definite status within the category of convex sets, namely they're the free ones, the free convex, devised to Hilbert space, a convex set, but a different kind of one. I've never actually seen much written about that, Ambrose, or Gleeson rather, which sort of extent of which that convex set is not.

50:00 Can you explain to me what a free convex, what the condition for a convex set to be a free convex set is. I'm afraid I don't know anything about convex sets. Actually, I'd better go in and get the, sorry, very, very basic question. Convexity, what is the condition of convexity?

52:30 Well, I was going to try to explain it, but there are many different things. Similar things, you could have different base categories and slightly different meanings to this. The crucial idea is that it's a space such that between any two points you have a straight, not an extended line, but just between two points. It's literally a convex, full convex object. It could be an ellipsoid or a square box or a solid tetrahedron or anything like that, you see. Right. So the free thing, so you have the underlying object where you forget this structure, the underlying set or the underlying topological space or whatever, and then the left adjoint to that. Yeah, I have to, if there isn't any sugar. That's fine, thanks. Yeah, thanks very much. So the left adjoint to the underlying is normally what's called free. Yeah. So that means, but I'm going to use the word P for that. So, the morphisms from P of X to C, which preserve these convex combinations, and the maps from X into C, or more exactly, the underlying of C, which are merely continuous, or in fact arbitrary maps in this case, which have nothing to do with this convex structure in other words, these are in natural one-to-one correspondence, this is the adjointness, or otherwise expressed. There is a junction map, delta, which is actually the Dirac delta in a way. In other words... So we're going to come back to that, of course. In other words, among the probability distributions on a space, there is the one that's concentrated

55:00 at a single point. It gives that point probability one, and any set that doesn't contain that point, probability zero. So that's what delta does. To each point, you assign that distribution. But this is somehow the space of all distributions on X. There's the associated monad, which is also called P, I guess. So it's the set of all probability distributions considered as a set or considered as a compact topological space or whatever you attribute to the base. Thank you. Yeah, okay, great. Thank you very much. So this bijection, if you have an arbitrary convex space C and an arbitrary map, I mean, in other words, it might be a continuous map if X calls for that, but it has nothing to do with convexity because X itself doesn't, then there's a unique convex map which extends it, a space of or set of reasonable probability measures, extensive quantities on X, free. Pre-convex. That's what this says, okay? Mm-hmm. Yeah. You see what I was saying about it back there? I hope the meaning of this symbol doesn't get lost or they won't understand us a hundred years from now. Because this little sign means all this. Yes. It does pack an awful lot of punch, doesn't it? Okay. Now, more precisely, a way of realizing this can be constructed in stages like this.

57:30 Real numbers, multiplication. You always have the category of them. Rig for us is commutative, by the way. It's a commutative multiplication. Non-commutative ones would be called R-algebras or something. So if you consider the modules over a rig, or otherwise you could call them R-linear spaces, Underlying and free for that, too. But it's a closed linear category. Closed means you have a tensor and a hom. You know what that... Yeah, yeah. ...which are adjoint to each other. Yeah, which are adjoint to one another. It's like lambda conversion, only not Cartesian. Yeah. And linear is this fact that products, finite products and coproducts agree. They're called byproducts. So actually what I'm going to say could be applied to any proposed linear category. So the thing is to consider their comma category, V over R. R itself is the unit object, tensor R, tensor V, isomorphic to V, or hom, R, V, also isomorphic to V. It's a particular object, and so you can consider this as comma category. So, so the thing is that this is one interpretation of convex. It depends on what kind of rig it is. This construction, if you applied it to a rig which was actually a ring, and it really, really had negatives, then this would not be convex but affine spaces, but so, so let's put a condition here, right, if we say that, if we say that A plus B equals zero implies...

1:00:00 This kind of very positively prevents it from having negatives, and so means that R is sort of positive. So in that case, it's reasonable to call these things convex rather than affine. But that's a different parenthesis. If instead, it would have been called affines, basically. So why this abstract business here? So the free module is in some sense the free, the extensive quantities. If V were free, if you think of V as being all measures on X, then the operation of taking the total value of a district, which gives you something in R itself, this would be an example of an object in the common category. Oh, I see. If there's a map from x to 1, so if you apply m sub r of x to this, m sub r of 1, of course, is always r. The free linear space with one basis element is a line which is r. So you get this induced map. So this is an example of something overall. It's not just an example of something in D.

1:02:30 The point being, if you consider the elements of this thing which map to one here, those are the distributions whose total is one. That's what a probability is. And since we put this condition, everything is positive. So the additional thing you define probability. A probability is an extensive quantity which is positive. ...and has total one in some sense, right? These are the two main conditions. So the positivity has been sort of forced by the very choice of the rig we work with, and having total one is somehow mirrored by working in this V over R because what's in work is a total structure of V and a total of W has to be preserved. So such an F... We'll map the things of total one to things of total one in particular. Right. So that the maps in the comma category are basically just maps of the probabilities. So in other words, this is one way of realizing the slight difference here. This functor acts as an inclusion, but it's not completely trivial. In other words, the measures considered with the total, it's a linear space equipped with this totality map to the reference point R.

1:05:00 So, in that way, the probability measures agree with M, except that M is conditioned by having this totality, by using the comma category instead of G. So, how am I telling you this? Well, you're filling me in an awful lot of detail about convex, how to think about convex faces. Yeah, that's right. This gives, in other words, this gives you a purely algebraic way to think about it. And you're going to explain to me what it is that's different in the, you know, the quantum case, well, the quantum probability measures. Well, they're already there, you see. I mean, the free ones are very special ones. Yeah, yeah. With algebras, every convex set. Did I make explicit, I mean, the usual thing about convex combinations? No, no. If you have a bunch of scalars that add up to one, and as I say, if we put this condition on R, that means that they're positive too. That's again the difference with the affine is that you don't require the lambdas to be positive. But if we set it up the way I did, we simply never mention things that aren't positive. It's more convenient that way to fix the condition all the time. And moreover, you have a corresponding family of distributions. They have the property that the total for each of those, since F is a linear map, V is the category of linear spaces, it commutes with this summation and multiplying by lambda. So these are usually called convex combinations. That is, linear combinations whose coefficients are positive, these are called convex combinations.

1:07:30 You know, the usual point of view of Neyman-Pearson statistics and so forth is that one-third times true plus thirds times false. It's a convex combination of real-life points, you see, just some kind of mixed point. It's also called a mixture. It's a convex combination or mixture of quantum mechanics or just convex combinations of pure states. Yeah, and they're the same thing. What, convex combinations of pure states and mixed states? Yes, that's by definition the same thing. Well, yeah. Yeah, if you look, yeah. I mean, the values of delta are pure states. That's right. I mean, the convex sets have one advantage over linear spaces. In the sense that if there is a basis it's unique. The linear space is completely arbitrary basis but if a convex set is going to be free, that is of the form p of x, you can extract the x already from it uniquely. It could consist of so-called extreme points. If p is in p of x and to say that p is delta of x for some x. This is equivalent to saying that it's pure, which means that if you express p as a convex combination of other things, where again, p to lambda i are positive, which, as I say, if you set it up correctly, you never have to mention that, but sum is one.

1:10:00 Yeah. So for all expressions of p as a mixture of other things, in fact, the mixture is trivial. There exists a unique i, such that lambda i is equals one, and hence the rest are zero. So any way of expressing p as a mixture is in fact trivially just picking out one of the things being mixed by letting the mixing coefficients be one and the rest zero. That defines a kind of, a very special kind of element p of any convex set, called an extreme point. And in the case of a free convex set, in other words, the space of probability measures, that's the same as being the value of delta. So another way of expressing freeness is that, yeah, the P, big P, C is equal to P , if and only if every element is uniquely a mixture of extreme points. And there's the so-called Krine-Muhlmann theorem, which says that much more general convex sets at least have the property that every element is a mixture of extreme points. There are enough extreme points, maybe not uniquely, but were uniquely expressible. But let's take some simple examples. Yeah, sure. So, for example, if x is, and p of x is, what would lead us to holding that interval? But if X itself is continuous, then P of X is infinite dimension. For an example of a convex set that's not free, take a solid square.

1:12:30 So here the convex set itself is finite dimensional, not infinite dimensional. But it's not free, because the only free ones are triangles. This is a one-dimensional triangle, a two-dimensional triangle, a tetrahedron, etc. The three convex sets are also called simplices, because they really are geometrical simplices, but this thing, of course, is a quotient of p of four, you can sort of imagine that if you take four elementary events, now consider all possible probability distributions on these four events, now that's a solid tetrahedron, that's a three-dimensional. Object which is a convex set by so so take your map four into okay this is c so four itself is obviously mapped into c then you choose these four points by the universal property of the left to join us yeah you get a unique map convex map maps that preserves that preserves mixtures from p or four from the solid tetrahedron three-dimensional tetrahedron I wish I had made the picture bigger. You have to kind of squash it, squash this thing in such a way that the four points remain sort of semi-independent, but somewhere in the middle. I mean, this line going across the front here... Well, use another bit of paper. This doesn't look very good. This is supposed to be a... that's even worse. We'll try again. Topology. Okay. Good. Now this will be... So there you see the three... Well, it's also a square. Yeah. Sure. I can see this. This is three-dimensional, though. Yeah. Yeah. But you squash it onto the actual square, which I can sort of draw the same way.

1:15:00 This is now... Yeah. This is three-dimensional. Three convex on four generators. This is a two-dimensional C convex. Not free. But it does have four extreme points, that's the point, you see. These are obviously the extreme points. In other words, if you, I mean, if you, a mixture of points is always in the convex interior of those points, so to speak. So, for example, this point could be expressed as a mixture of these. These cannot be expressed as mixtures of other points and strays. So these are exactly the extreme points. And the extreme points span the thing, but not uniquely, because the thing that's spanned uniquely or freely by four points is a three-dimensional tetrahedron. Extreme points but not uniquely so because you can see right here that I mean this whole three-dimensional thing is being squashed in a linear manner in a way that preserves convex combinations the obvious example is that well this point on the front edge and this and that point back there on the back edge which are different here have become the same over here yeah so it's not uniquely expressed they're too different Convex combinations of three, sorry, four, which in the square represents the same thing, obviously, and in the middle, at least one point, you can figure out how many there are and so on, but there's an obvious example at some point in the middle, which is non-unique. So that's the kind of thing, in the finite case, that's going on here. But now, so there, but every convex set is an image, it's your objective.

1:17:30 Mixture preserving, or convex, or affine, that. Well, in fact, you see, I always thought, I always wondered why David Berm gave up. Why David? Berm. Yeah. Gave up. Because this already says, you see, that anything... Yeah. This is a classical system. And it contains C. Yeah. I mean, it's just, it's got by, you have to, you have to induce a lot of relations. Yeah. I mean, this is already a simple case. I mean, this will never be equal to p of x, but it is p of x modulo a certain congruence relation, which one could keep track of. Yeah, yeah. So, I mean, so these x, these x is the hidden variables or something. Yeah, yeah. Yeah, I see the connection. Well, yes, I have to try and understand more about how Heide does think about these non-local hidden variables. I don't know if there's a connection with simplicial topology at all, and it seems to be in terms of... Well, not simplicial topology, because, you see, these things are all convex, right? Oh, right, yes, so it's not... The idea of simplicial topology is that you make geometric, you make models of very non-trivial, homologically non-trivial spaces by thinking of them as convex pieces that are glued together. So the pieces are trivial. Yeah, so here we're really just feeling... ...homological point of view, but it's the structure of the interlocking of these pieces which gives rise to the homology. Yes, right, where at this point we're simply dealing with the simplicities as convex. Right, with the objects which from the point of view of algebraic topology are all trivial. Yeah, yeah, yeah. But they do have, of course, a lot of subtlety to them. How much I never realized. Well, no, I mean, actually, as I say, my student's thesis is really the only one I know of which systematically talks about this stuff at all. She applies it to things like statistical decision theory and dynamic programming and stochastic processes.

1:20:00 Well, I was going to ask you a bit about that. I mean, how does it, and you mentioned Gleason's theorem. How's that entering? And how do these convex sense weigh in? The thing is that, for example, an important example of C would be what are called probabilities, but on the unit sphere in Hilbert space. But in other words, instead of having the... Usual additivity, if you have two disjoint subsets of events and you take their union, and the probability of the union should be the sum of the probabilities, where disjointness just means the intersection is zero, and these were subsets of something. Well, instead, if you have orthogonality structures, you put the same condition. If A and B are things that are orthogonal, then their probabilities should add up. But that's not at all the same thing as an infection zero in a sort of Boolean or hiding context. No, no. So, but it does give rise to a convex set, you see, but not one of the, not something that I'm calling, these are p of x in the sense of, roughly speaking, and they're used in the sense of, you know, additive set functions, you see. Yeah, yeah, yeah. There'll be the kind of thing, and there may be small technical differences, but conceptually it's the same kind of thing as the additive set functions, but the things that are additive on the subspaces of Hilbert space, additive with respect to the orthogonality relation, is a convex set. So it's an object in this category, but it's very much not of the form p of x. It's a quotient of p of x. It's a general fact. But Gleason's theorem is sort of going in the other way, you know, it's a property that this particular example has, which shows how much, how really far different it is from a free-convex set.

1:22:30 Yeah. Perhaps we can look up the actual statement in here. Well, I've got a, yeah, actually, Helzworth in that survey of quantum logic does have a fairly full statement of Gleason's theorem. Well, in fact, the original paper is reprinted in that book on... So that's, well, just for the purpose of summarizing this, this, so therefore we have a contradiction. Why didn't Berm persist? Well, well, more to the point, why didn't, yes, why didn't Heine work on this road instead of the way, well, I think because he was already going down an algebraic geometry route in terms of differential forms, but, which I, the thing I was saying I think might be an instance of. This is somewhat interesting. In Mackey's Foundation of Quantum Mechanics, there is this idea about convex sets. Not quite so simple and abstract, but still, it's clear from his axioms that the crucial point is some kind of convex set of states, and that observables are functions of some sort defined on... This convex set could never be p of x without some relation, but you see, this says it's p of x modulo, let's make this more precise even, there's an x and a y in a pair of maps, so that this is a co-equalizer, so it's a free kind, a classical statistical system, modulo another one. Well, you impose relations. And, of course, these maps could be thought of as ordinary maps from y into p of x, that is, random maps, two random maps from x from y to x, and you co-equalize those into convex sets, and that's the arbitrary convex sets, including this example, which is considered central in quantum mechanics, which is highly non-free in its own right.

1:25:00 I mean, abstractly, this is just like, say, group theory, you know, those free groups, which are very, very special, but an arbitrary group is a quotient of a free group, modulo another free group. Yes, I see. Due to the influence of Lebesgue and Karnaker, the countable additivity is considered very important. Yeah, I was going to ask you, because it's not in fact going off the subject as much as all that, about your remark about Lebesgue integration. Sorry, it's not... my sleep deficiency is catching up with me a bit. Chronic or Lebesgue. Yeah, Chronic or Lebesgue, as opposed to Piano. Who's the other guy? Oh, Grassmann? No. Well, you didn't mention Grassmann. Oh, yeah, yeah, Kurzweil and Hitchcock. That's right. Can you explain that to me, the difference between the approaches? That was one of my sort of six top questions I wanted to ask you about. It was about number five. In all of industrial processes, there arise differential equations. Non-autonomous means that the T occurs there, where F is not Lipschitz, let's say. So, I mean, the usual elementary theory of differential equations is based on the assumption that F is Lipschitz, so that one can apply various simple-minded methods of Pixpoints and so forth to get existence of solutions to differential equations. But, in actual practice, that may not be Lipschitz. So one needs a more subtle kind of integration theory in order to, and so Kurzweil developed this notion of integration, which then Hinstock, I think is British actually, perhaps Scottish. I forget the dates a little bit later. I guess Scottish is rooted in British nowadays. And then later on even...

1:27:30 Even the great American integration theorist, McShane, wrote a book called Unified Integration Theory, intending to include not only this, but those topastic entities, and then a Keras monograph, claiming to explain this, as all the MAA publications, the pretext of explaining, but actually obfuscates their systematic methods since 1893, Keras being merely the... Spiritual, Godfather, the whole thing. But anyway, I mean, it does have some... This is called the generalized Riemann integral. The basic point is this, that suppose you have a measure mu and a function f, and that the integral fd mu in the space x equals a certain value L.

1:30:00 So that they're all partitions of x, size less than delta, the elementary interval less than epsilon. The whole Riemann's early notes, this is a story like Lakotash and John Cleve, you see, about Cauchy. So I don't know if it's valid or not, but back to the, I should tell you who told me about this, maybe you know him, his name is Gerald Goodman, he's an American, lives and works in Florence, lives in a marvelous 12th century tower, hard times, but he's the one who first explained this to me. I think he's the first one to have proved the fundamental theorem of calculus in this context. Please do. It's extremely interesting. It's extremely natural for the fundamental theorem of calculus. Anyway, the historical quibble. Maybe Riemann meant the right thing all along. It's just that he was misinterpreted later, or maybe he made a mistake himself later when he wrote it up more precisely. It has to do with what do we mean by delta, by partition? Two things. Delta is a constant and a partition is countable. In other words, the AI is a family of sets whose sum is the whole space, but of course, they're measurable sets, whatever that means. I mean, in a simple case, you can just take intervals. This is not really a big deal, but it's a countable number of them in principle.

1:32:30 Of course, in either case, a partition involves a family of sets and a family of chosen points in each of the sets, so you evaluate the function at the chosen point, you multiply that by the measure of the piece that it's in, and you just have this sort of, right, and of course they should be disjoint, I mean, I suppose the natural thing is to say that the measure of the intersection is zero if you have two different ones. Or you could say that they're literally disjoint. That's not, again, that's not the important thing. The thing is, delta is just a constant number. Partition is countable. And size, well, size means that the soup over I of me will weigh. So the size, the largest one of these should be less than this constant value, whereas the Kurzweil, Henstock, etc. Actually, McShane generalizes this, the point, the sample point might sort of just be on the edge of the set and not really in it and stuff like this, but again I don't think that's, at least Goodman and I agreed that that didn't seem to be a good thing to make that generalization. The first is arbitrary, so you make epsilon very small, you need to take very many small pieces and so on, but finite, no mysterious actual infinity at any given stage of the approximation, whereas according to Lebesgue, even at the first stage, you might have a countable infinity.

1:35:00 So this is, from a point of view of numerical analysis, this is much more rational. Everything is actually... ...finite when it comes to approximating and doing calculations, and then the definition of size, well, no problem, see the size of the partition, you have this finite family of sample points contained in a finite family of disjoint measurable sets which add up to the, well, I mean, to say that the size is less than delta, which is the actual cause that occurs in the definition, is simply to say that the measure of the... The i set is smaller than the function delta applied to the sample point. Now, what you get is that if f is Lebesgue integral. Now notice that already Lebesgue has a trick here because this by definition means that not f but absolute value of f is integral. Now you sense that they do not require that. F is, in fact, not conversely, so that the finite partitions, functional delta, is actually more general, and more functions can be integrated. Very striking. Very. And the other thing is the fundamental theorem of calculus.

1:37:30 Yes, I was going to ask you about that. You see, again, Lebesgue and his followers have some version of the fundamental theorem, of course, but it has hypotheses. The true fundamental theorem of calculus has no hypotheses. Fundamental theorem without hypotheses is that, if I can get this right, that if big F has a derivative, that's the only assumption, one which is necessary to even make sense of the formula, the derivative F implies that F equals the integral of little f, the usual. The vague version has additional strange hypotheses due to the unnatural assumption that delta is constant and due to the business about the vague. Why do I say this is very natural? Because the definition of derivative is local and there's no uniformity of the delta in the definition of derivative, the function of derivative. So you have some story of epsilons and deltas, but at each point, independently, more or less than, you know, relatively independently, so naturally your delta is a function. So you just sort of substitute that back in and read that formula backwards, you have this theorem. There's no, because this is defined to mean that the delta is allowed to be, the usual, I mean, as a special example of this implication, the famous, the function which is... 0 and 1 according to whether the point is rational or irrational. An example I always like to quote to show how great Lebesgue integration is, classical Riemann, where classical Riemann means the constant delta, not that anyone would ever want to use this function in real life, but it's the one that they put great sore in. It's easily seen to be integral in a sense. It's just a point that the function delta has to be chosen in a very clever way.

1:40:00 The A is one way at rational and the other way at irrationals, but the partitions can be finite. You see, intuitively the idea is that if you're going to integrate, if you're going to achieve this approximation, if f is very, very highly variable in a certain region, then you're going to have to take very small partition pieces there, and the function delta is going to have to be quite variable there as well, whereas in a part where f is more slowly varying, You can afford to use quite coarse partitions and quite relatively constant delta. But all these possibilities are allowed, and so even this monstrosity can be easily accounted for. Near irrational, you add a rational, you do one thing with the delta, and it's irrational to do another thing. Likewise for the sizes of the partitions. I mean, that's all explained in McLeod's book. Right. Right. Whereas, of course, in the Riemann version, you really couldn't handle this. That's right. That's what I'm saying. That's the example that's always mentioned, first of all, is how helplessly backward this Riemann integral is. It can't deal with these important functions that come up in everyday life. Well, it's actually... Of course, it's exactly the sort of thing that Lakatos lent his support to by all this... ...stuff about monster barring, the way that one stretched concepts was in order to... The monster barring was the most important way in which concepts got stretched in the history of mathematics. You know, this stuff that's in Proust's reputation. Oh, you mentioned it. In fact, I think he actually uses this specific example, Riemann and Lebesgue integration in some of his papers, which is really, a lot of it is seen when you look at it really carefully like this. It seems to be a very hasty and superficial history of mathematics.

1:42:30 Right, right. I mean, this I could mention. Well, I mean, in other words, all of these things that are produced by bad infinity are the examples that are constantly cited as things that you must have to be reasonable. Yes. You must have this function integrable. You must have the piano curve spilling space. You must have... These things are really... What's his theory? I mean, I would bar these monsters by simply working in a category where they don't exist, but that's not his... Well, no, I don't think that... The trouble is monster barring is supposed to be an ad hoc procedure. Once you stretch your concepts, the monsters no longer need to be barred because they fall under the concepts once they've been satisfactorily stretched to accommodate the new examples. This is very helpful to me now. I've said a lot more of that. I always thought that the Riemann Theories sort of intuitively seemed a lot... Yeah, well, the finite partitions, as I say, it's a thing which anyone would use in actual numerical analysis. So, yeah, Isakrusen's definition of derivative at each point x gives non-constant delta in any case. Yeah, exactly. Well, thanks. That's helped me a lot. Now, the next thing I really wanted to talk about, I guess, is to go back to... The intensive, extensive quantities pulling forward into the CCRs. But I think my sleep deficit has caught up with me a bit. I'd like to go and just score some crash for about an hour. Can I do that? And then, because Jerry Ketcheran is coming at 6, between 6 and 6.30, which is now 4.30. So if I can just crash out for an hour. And if you want to, incidentally, Peter will have plenty more cigarettes because he's... The only man in the world I know that churns smokes at least three times more than you do, so if you want to get some, is that okay if I can churn it for a bit, and then, can I take those? Oh yeah, okay, yeah, and then if I can just, as I say, get a bit of rest and then I'll be fit for talking this evening, and I've got the papers, the notes of the intensive extensive quantitative CCRs.

1:45:00 And I've got the list of the other things I wanted to ask you. And I'll try and dig the tape out too before we go out tonight, just so I've got it there to copy in the morning. Okay, anywhere else? If you want to get some more tea or anything, I'll just watch a bit of television. Behind, Greg. On the contrary, it's very nice. Very nice having you both. Did you get any cigarettes? Yeah, yeah, he's staying here last night. He's staying here tonight. He's going back to the States tomorrow. I'm just going to go and crash out for an hour or so because we didn't get any sleep at all last night. Well, he did, but I didn't. Because we came down from the Oslo-Milk train from Bangor. So I'm just going to crash out for an hour or so because the electric chair is coming around about six o'clock. Oh, well, he might be here a bit early actually. Well, he'll give you a lift back in. Yeah, sure. That's one of the things I wanted to do. But you really have sorted out the business with Biff because it sounded very bad. What happened? Well, you see, I don't remember the situation. Yeah, yeah, yeah. Well, the situation is that it's okay. But there is a problem with this guy in the States that hasn't produced the tickets. You know that? Yeah. I mean, have they got the tickets now? No, they haven't. Well, that means Witty's down the tube, doesn't it? Oh, well, goodbye, Witty. Well, he won't be giving me anything at all, but all the same. Oh, come on, Witty. OK, we don't have to talk about that now.